Random walks on planar crystallographic groups (Q2487377)

The authors consider a crystallographic group \(G=pgg\) generated by two perpendicular gliding reflections \textit{O} and \textit{P} and the following code: \((P*O)^2=(P^{-1}*O)^2=E.\) They prove the following main result: A random walk on the graph of the group \(G\) has the property of return if and only if its transition probabilities satisfy the conditions \(p_1>0\) and \(p_2>0\). Moreover for the group \(G\) the authors calculate the generating matrix and reduce the property of random walks on planar crystallographic groups to the local behavior of the resolvent.